Symmetry in Nonlinear Mathematical Physics - 2009
Dmytro Popovych (National Taras Shevchenko University of Kyiv, Ukraine)
Generalized IW-contractions of Lie algebras
We prove that there exists just one pair of complex four-dimensional Lie algebras
such that a well-defined contraction among them is not equivalent to a generalized IW-contraction
(or to a one-parametric subgroup degeneration in conventional algebraic terms).
Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra.
The example we constructed demonstrates that even in the dimension four generalized IW-contractions
are not sufficient for realizing all possible contractions,
and this is the lowest dimension in which generalized IW-contractions are not universal.
Moreover, this is also the first example of nonexistence of generalized IW-contraction
for the case when the contracted algebra is not characteristically nilpotent
and, therefore, admits nontrivial diagonal derivations.
The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient
to realize all generalized IW-contractions of four-dimensional Lie algebras is found.
We also present a simple and rigorous proof of the known claim
that any diagonal contraction (e.g., a generalized IW-contraction) is equivalent to
a generalized IW-contraction with integer parameter powers.